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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Nonuniform sampling ： ウィキペディア英語版 Nonuniform sampling Nonuniform sampling is a branch of sampling theory involving results related to the Nyquist–Shannon sampling theorem. Nonuniform sampling is based on Lagrange interpolation and the relationship between itself and the (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker–Shannon–Kotelnikov (WSK) sampling theorem. The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition.〔Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000〕 Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction. The general theory for non-baseband and nonuniform samples was developed in 1967 by Henry Landau.〔H. J. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math., vol. 117, pp. 37–52, Feb. 1967.〕 He proved that, to paraphrase roughly, the average sampling rate (uniform or otherwise) must be twice the ''occupied'' bandwidth of the signal, assuming it is ''a priori'' known what portion of the spectrum was occupied. In the late 1990s, this work was partially extended to cover signals for which the amount of occupied bandwidth was known, but the actual occupied portion of the spectrum was unknown.〔see, e.g., P. Feng, “Universal minimum-rate sampling and spectrum-blind reconstruction for multiband signals,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, 1997.〕 In the 2000s, a complete theory was developed (see the section Beyond Nyquist below) using compressed sensing. In particular, the theory, using signal processing language, is described in this 2009 paper.〔(Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals ), Moshe Mishali and Yonina C. Eldar, in IEEE Trans. Signal Processing, March 2009, Vol 57 Issue 3〕 They show, among other things, that if the frequency locations are unknown, then it is necessary to sample at least at twice the Nyquist criteria; in other words, you must pay at least a factor of 2 for not knowing the location of the spectrum. Note that minimum sampling requirements do not necessarily guarantee numerical stability. ==Lagrange (polynomial) interpolation== For a given function, it is possible to construct a polynomial of degree ''n'' which has the same value with the function at ''n'' + 1 points.〔Marvasti 2001, p. 124.〕 Let the ''n'' + 1 points to be $z\_0,\; z\_1,\; \backslash ldots\; ,\; z\_n$, and the ''n'' + 1 values to be $w\_0,\; w\_1,\; \backslash ldots,\; w\_n$. In this way, there exists a unique polynomial $p\_n(z)$ such that :$p\_n(z\_i)\; =\; w\_i,\; \backslash texti\; =\; 0,\; 1,\; \backslash ldots,\; n.$〔Marvasti 2001, pp. 124–125.〕 Furthermore, it is possible to simplify the representation of $p\_n(z)$ using the interpolating polynomials of Lagrange interpolation: :$I\_k(z)\; =\; \backslash frac)\backslash cdots(z-z\_n)\})\backslash cdots(z\_k-z\_n)\}$〔Marvasti 2001, p. 126.〕 From the above equation: :$$ I_k(z_j) = \delta_ = \begin 0, & \textk\ne j \\ 1, & \textk = j \end As a result, :$p\_n(z)\; =\; \backslash sum\_^n\; w\_kI\_k(z)$ :$p\_n(z\_j)\; =\; w\_j,\; j\; =\; 0,\; 1,\; \backslash ldots,\; n$ To make the polynomial form more useful: :$G\_n(z)\; =\; (z-z\_0)(z-z\_1)\backslash cdots(z-z\_n)$ In that way, the Lagrange Interpolation Formula appears: :$p\_n(z)\; =\; \backslash sum\_^n\; w\_k\backslash frac$〔Marvasti 2001, p. 127.〕 Note that if $f(z\_j)=p\_n(z\_j),\; j=0,\; 1,\; \backslash ldots,\; n,$, then the above formula becomes: :$f(z)\; =\; \backslash sum\_^n\; f(z\_k)\backslash frac$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nonuniform sampling」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.